Differential equation and inequalities of the generalized k-Bessel functions

In this paper, we introduce and study a generalization of the k-Bessel function of order ν given by $$ \mathtt{W}^{\mathtt{k}}_{\nu , c}(x):= \sum_{r=0}^{\infty } \frac{(-c)^{r}}{\Gamma_{\mathtt{k}} ( r \mathtt{k} +\nu +\mathtt{k} ) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}}. $$ We also indicate some representation formulae for the function introduced. Further, we show that the function $\mathtt{W}^{ \mathtt{k}}_{\nu , c}$ is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function $\mathtt{W}^{\mathtt{k}} _{\nu , c}$ , particularly, in the case $c=-1$ . We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of $\mathtt{W}^{ \mathtt{k}}_{\nu , c}$ .